Local Asymptotics for the Time of First Return to the Origin
نویسندگان
چکیده
We consider a transient random walk on Z which is asymptotically stable, without centering, in a sense which allows di¤erent norming for each component. The paper is devoted to the asymptotics of the probability of the rst return to the origin of such a random walk at time n. 60G50 multidimensional random walk, transience, rst return to the origin, local limit theorem, defective renewal function, locally subexponential distributions, Banach algebra of distributions 1 Introduction Let Sn, n 0, be a random walk in Z generated by independent identically distributed steps n = ( n1; : : : ; nd), n 1, that is, S0 = 0, Sn = 1+ : : :+ n. Denote 0 = 0 and recursively n+1 = minfk > n : Sk = 0g; by standard convention min ; = 1. Then = 1 is the rst return to the origin of the random walk Sn. In this paper we study the asymptotic behaviour of pn := Pf = ng = Pfthe rst return to zero occurs at time ng as n!1. Put G(B) = Pf 2 Bg and
منابع مشابه
Local asymptotics for the time of first return to the origin of transient random walk
We consider a transient random walk on Z which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time n.
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